Geometrical aspects of Schlesinger's equation

✍ Scribed by Nigel Hitchin

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 822 KB
• Volume: 23
• Category: Article
• ISSN: 0393-0440
• DOI: 10.1016/s0393-0440(97)80005-8

✦ Synopsis

The equation (Schlesinger's equation) for the isomonodromic deformations of an SL(2, @) connection with four simple poles on the projective line is shown to describe a holomorphic projective structure on a surface. The space of geodesics of this structure is, by a primitive version of twistor theory, a two-dimensional complex Poisson manifold containing complete rational curves. The Poisson structure degenerates on a divisor and it is shown that the complement of the divisor is a symplectic manifold which can be identified with the quotient of the moduli space of representations of a free group on three generators in SL(2, c) by the action of a braid group.